Chapter 6 Probability PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau

Chapter 6 Learning Outcomes 1 Understand definition of probability 2 Explain assumptions of random sampling 3 Use unit normal table to find probabilities 4 Use unit normal table to find scores for given proportion 5 Find percentiles and percentile rank in normal distribution

Tools You Will Need Proportions (Math Review, Appendix A) Fractions Decimals Percentages Basic algebra (Math Review, Appendix A) z-scores (Chapter 5)

6.1 Introduction to Probability Research begins with a question about an entire population. Actual research is conducted using a sample. Inferential statistics use sample data to answer questions about the population Relationships between samples and populations are defined in terms of probability

Figure 6.1 Role of probability in inferential statistics FIGURE 6.1 The role of probability in inferential statistics. Probability is used to predict what kind of samples are likely to be obtained from a population. Thus probability establishes a connection between samples and populations. Inferential statistics rely on this connection when they use sample data as the basis for making conclusions about populations.

Definition of Probability Several different outcomes are possible The probability of any specific outcome is a fraction or proportion of all possible outcomes

Probability Notation p is the symbol for “probability” Probability of some specific outcome is specified by p(event) So the probability of drawing a red ace from a standard deck of playing cards could be symbolized as p(red ace) Probabilities are always proportions p(red ace) = 2/52 ≈ 0.03846 (proportion is 2 red aces out of 52 cards)

(Independent) Random Sampling A process or procedure used to draw samples Required for our definition of probability to be accurate The “Independent” modifier is generally left off, so it becomes “random sampling”

Definition of Random Sample A sample produced by a process that assures: Each individual in the population has an equal chance of being selected Probability of being selected stays constant from one selection to the next when more than one individual is selected Requires sampling with replacement

Probability and Frequency Distributions Probability usually involves population of scores that can be displayed in a frequency distribution graph Different portions of the graph represent portions of the population Proportions and probabilities are equivalent A particular portion of the graph corresponds to a particular probability in the population

Figure 6.2 Population Frequency Distribution Histogram FIGURE 6.2 A frequency distribution histogram for a population that consists of N = 10 scores. The shaded part of the figure indicates the portion of the whole population that corresponds to scores greater than X = 4. The shaded portion is two-tenths (p = 2/10) of the whole distribution.

Learning Check p = 1/52 p = 12/52 p = 3/52 A deck of 52 cards contains 12 royalty cards. If you randomly select a card from the deck, what is the probability of obtaining a royalty card? A p = 1/52 B p = 12/52 C p = 3/52 D p = 4/52

Learning Check - Answer A deck of 52 cards contains 12 royalty cards. If you randomly select a card from the deck, what is the probability of obtaining a royalty card? A p = 1/52 B p = 12/52 C p = 3/52 D p = 4/52

Learning Check TF Decide if each of the following statements is True or False. T/F Choosing random individuals who walk by yields a random sample Probability predicts what kind of population is likely to be obtained

Learning Check - Answers False Not all individuals walk by, so not all have an equal chance of being selected for the sample The population is given. Probability predicts what a sample is likely to be like

6.2 Probability and the Normal Distribution Normal distribution is a common shape Symmetrical Highest frequency in the middle Frequencies taper off towards the extremes Defined by an equation Can be described by the proportions of area contained in each section. z-scores are used to identify sections

Figure 6.3 The Normal Distribution FIGURE 6.3 The normal distribution. The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (frequency). The equation is provided on the slide. (Pi and e are mathematical constants.) In simpler terms the normal distribution is symmetrical with a single mode in the middle. The frequency tapers off as you move farther from the middle in either direction.

Figure 6.4 Normal Distribution with z-scores FIGURE 6.4 The normal distribution following a z-score transformation.

Characteristics of the Normal Distribution Sections on the left side of the distribution have the same area as corresponding sections on the right Because z-scores define the sections, the proportions of area apply to any normal distribution Regardless of the mean Regardless of the standard deviation

Figure 6.5 Distribution for Example 6.2 FIGURE 6.5 The distribution of SAT scores described in Example 6.2.

The Unit Normal Table The proportion for only a few z-scores can be shown graphically The complete listing of z-scores and proportions is provided in the unit normal table Unit Normal Table is provided in Appendix B, Table B.1

Figure 6.6 Portion of the Unit Normal Table FIGURE 6.6 A portion of the unit normal table. This table lists proportions of the normal distribution corresponding to each z-score value. Column A of the table lists z-scores. Column B lists the proportion in the body of the normal distribution up to the z-score value. Column C lists the proportion of the normal distribution that is located in the tail of the distribution beyond the z-score value. Column D lists the proportion between the mean and the z-score value.

Figure 6.7 Proportions Corresponding to z = ±0.25 FIGURE 6.7 Proportions of a normal distribution corresponding to z = +0.25 (a) and -0.25 (b).

Probability/Proportion & z-scores Unit normal table lists relationships between z-score locations and proportions in a normal distribution If you know the z-score, you can look up the corresponding proportion If you know the proportion, you can use the table to find a specific z-score location Probability is equivalent to proportion

Figure 6.8 Distributions: Examples 6.3a—6.3c FIGURE 6.8 The distribution for Example 6.3a—6.3c

Figure 6.9 Distributions: Examples 6.4a—6.4b FIGURE 6.9 The distributions for Examples 6.4a and 6.4b.

Learning Check Find the proportion of the normal curve that corresponds to z > 1.50 A p = 0.9332 B p = 0.5000 C p = 0.4332 D p = 0.0668

Learning Check - Answer Find the proportion of the normal curve that corresponds to z > 1.50 A p = 0.9332 B p = 0.5000 C p = 0.4332 D p = 0.0668

Learning Check Decide if each of the following statements is True or False. T/F For any negative z-score, the tail will be on the right hand side If you know the probability, you can find the corresponding z-score

Learning Check - Answer False For negative z-scores the tail will always be on the left side True First find the proportion in the appropriate column then read the z-score from the left column

6.3 Probabilities/Proportions for Normally Distributed Scores The probabilities given in the Unit Normal Table will be accurate only for normally distributed scores so the shape of the distribution should be verified before using it. For normally distributed scores Transform the X scores (values) into z-scores Look up the proportions corresponding to the z-score values.

Figure 6.10 Distribution of IQ scores FIGURE 6.10 The distribution of IQ scores. The problem is to find the probability or proportion of the distribution corresponding to scores less than 120.

Figure 6.11 Example 6.6 Distribution FIGURE 6.11 The distribution for Example 6.6.

Box 6.1 Percentile ranks Percentile rank is the percentage of individuals in the distribution who have scores that are less than or equal to the specific score. Probability questions can be rephrased as percentile rank questions.

Figure 6.12 Example 6.7 Distribution FIGURE 6.12 The distribution for Example 6.7.

Figure 6.13 Determining Normal Distribution Probabilities/Proportions FIGURE 6.13 Determining probabilities of proportions for a normal distribution is shown as a two-step process with z-scores as an intermediate stop along the way. Note that you cannot move directly along the dashed line between X values and probabilities or proportions. Instead, you must follow the solid lines around the corner.

Figure 6.14 Commuting Time Distribution FIGURE 6.14 The distribution of commuting time for American workers. The problem is to find the score that separates the highest 10% of commuting times from the rest.

Figure 6.15 Commuting Time Distribution FIGURE 6.15 The distribution of commuting times for American workers. The problem is to find the middle 90% of the distribution.

Learning Check Membership in MENSA requires a score of 130 on the Stanford-Binet 5 IQ test, which has μ = 100 and σ = 15. What proportion of the population qualifies for MENSA? A p = 0.0228 B p = 0.9772 C p = 0.4772 D p = 0.0456

Learning Check - Answer Membership in MENSA requires a score of 130 on the Stanford-Binet 5 IQ test, which has μ = 100 and σ = 15. What proportion of the population qualifies for MENSA? A p = 0.0228 B p = 0.9772 C p = 0.4772 D p = 0.0456

Learning Check Decide if each of the following statements is True or False. T/F It is possible to find the X score corresponding to a percentile rank in a normal distribution If you know a z-score you can find the probability of obtaining that z-score in a distribution of any shape

Learning Check - Answer True Find the z-score for the percentile rank, then transform it to X False If a distribution is skewed the probability shown in the unit normal table will not be accurate

6.4 Looking Ahead to Inferential Statistics Many research situations begin with a population that forms a normal distribution A random sample is selected and receives a treatment, to evaluate the treatment Probability is used to decide whether the treated sample is “noticeably different” from the population

Figure 6.16 Research Study Conceptualization FIGURE 6.16 A diagram of a research study. A sample is selected from the populations and receives a treatment. The goal is to determine whether the treatment has an effect.

Figure 6.17 Research Study Conceptualization FIGURE 6.17 Using probability to evaluate a treatment effect. Values that are extremely unlikely to be obtained form the original population are viewed as evidence of a treatment effect.

Figure 6.18 Demonstration 6.1 FIGURE 6.18 A sketch of the distribution for Demonstration 6.1.

Any Questions? Concepts? Equations?